Cheban D.N. Global Attractors Of Non-autonomous Dissipative Dynamical Systems

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Method of Lyapunov functions 187

Proof. In the same way as in Theorem 5.9 we can show that the solution ϕ(·, u, f

)

of the equation (5.10) (g = f

) passing through u as t = 0 is extendable to the

right onto the whole semi-axis R

. Assume Y := Σ

and by (Y, R

, σ) denote

the dynamical system deﬁned by the following rule: π

(u, f

) := (ϕ(t, u, f

), f

τ+t

)

for all τ ∈ R, u ∈ E

and t ∈ R

. Deﬁne on X

:= E

× Y a function V

by the equality V(u, f

) := V (τ, u). Then V(π

(u, f

)) = V(ϕ(t, u, f

), f

t+τ

) =

V (t + τ, ϕ(t, u, f

)) = V (s, ϕ(s − τ, u, f

)) = V (s, x(s; u, τ)) (s = t + τ) and,

consequently,

(u, f

) = lim

t↓0

sup t

−1

[V(π

(u, f

)) − V(u, f

)]

= lim

s−τ↓0

sup(s − τ )

−1

[V (s, x(s; u, τ)) − V (τ, x[τ; u, τ))]

(τ, x(τ; u, τ)) =

(τ, u) ≤ −c(|u|) (|u| ≥ r).

According to Theorem 5.3 the non-autonomous dynamical system



(X, R

, π),

(Y, R

, σ), h



(h = pr

: X → Y ) is dissipative and, consequently, there exist R > 0

and l(u, τ) > 0 such that |ϕ(t, u, f

)| < R for all u ∈ E

, τ ∈ R and t ≥ l(u, τ ). As

it is in

[

137

]

we can show that the number l(u, τ) > 0 can be chosen not depending

on τ ∈ R. Reasoning in the same way that in the proof of Lemma 3.1, we obtain

that the equation (5.9) is dissipative. 

Note that Theorem 5.12 makes precise the known theorem of T. Yoshizava

[

137

]

[

325

]

–

[

326

]

5.3 Theorem of Barbashin–Krasovskii for non-autonomous dynamical

systems

The ideas and methods that we applied studying dissipative system in ﬁrst two para-

graphs of Chapter 5 allow formulating and proving of a series of statements about

the stability of non-autonomous dynamical system and, in particular, generalizing

of the known theorem of Barbashin–Krasovskii

[

]

on asymptotic stability.

Let (X, h, Y ) be a ﬁnite-dimensional vectorial bundle ﬁber, h(X, T

, π),

(Y, T

, σ), hi be non-autonomous dynamical system, θ

∈ X

:= h

−1

(y) be a null

element (|θ

| = 0) and Θ := {θ

| y ∈ Y } be a null section of the vectorial bun-

dle ﬁber (X, h, Y ). Below we will suppose that the null section Θ is invariant, i.e.

Θ ⊆ X is an invariant set of the dynamical system (X, T

, π).

Deﬁnition 5.3 the null section Θ is called uniformly stable if for every ε > 0

there exists δ(ε) > 0 such that y ∈ Y, x ∈ X

and |x| < δ implies |xt| < ε for all

t ≥ 0.

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188 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Deﬁnition 5.4 If Θ is uniformly stable and lim

t→+∞

|xt| = 0 for all x ∈ X, then

the null section is called globally uniformly asymptotically stable.

Theorem 5.13 Let Y be compact. For the null section Θ to be globally uniformly

asymptotically stable it is necessary and suﬃcient that there would exist a continuous

function V : X → R

satisfying the following conditions:

1. V (x) ≥ a(|x|) for all x ∈ X, V (θ

) = 0 for all y ∈ Y and Im a = Im V, where

a ∈ A.

2. V (xt) ≤ V (x) for all x ∈ X and t ≥ 0.

3. the lines of level of V do not contain non-null ω-limit points of the dynamical

system (X, T, π).

Proof. Suﬃciency. Let the conditions of the theory be satisﬁed. Show that the

null section Θ is uniformly stable. Suppose that it is not true. Then there exist

> 0, |x

| < δ, δ

↓ 0 and t

≥ 0 such that

| ≥ ε

. (5.29)

On the other hand, 0 ≤ a(|x

|) ≤ V (x

) ≤ V (x

) → 0 as n → +∞ and,

consequently, |x

| → 0. The last contradicts to the equality (5.29).

Now let us show that lim

t→+∞

|xt| = 0 for all x ∈ X. Really, if we suppose the

contrary then there exists x

∈ X (|x

| 6= 0) such that lim sup

t→+∞

t| > 0, i.e. there

exist ε

> 0 and t

→ +∞ for which

| ≥ ε

. (5.30)

Note that Σ

is relatively compact. In fact, a(|x

t|) ≤ V (x

t) ≤ V (x

) and,

consequently, |x

t| ≤ a

−1

(V (x

)) for all t ≥ 0. So, the sequence {x

} can be

considered convergent. Assume ˜x := lim

n→+∞

, then ˜x ∈ ω

. Reasoning in the

same way that in Theorem 5.1 we can show that there exists c ≥ 0 for which

V (x) = c for all x ∈ ω

. Since ˜x ∈ ω

and ˜x = lim

n→+∞

, from (5.30) follows

that |˜x| > 0, hence c = V (˜x) ≥ a(|˜x|) > 0. So, the lines of level of the function V

contain non-null ω–limit points. This contradicts to the conditions of the theorem.

The global uniform asymptotic stability of the null section is proved.

Necessity. Let the null section Θ be globally uniformly asymptotically stable.

Let V : X → R

be the function, deﬁned by the equality (5.1). Directly from the

deﬁnition of V follows that it satisﬁes to the conditions 1. and 2. of the theorem.

We will show that it is continuous. For that we note that under conditions of

Theorem the non-autonomous system



(X, T

, π), (Y, T

, σ), h



is dissipative. Now

we will show that J ⊆ Θ, where J is the center of Levinson of (X, T

, π). Indeed, if

x ∈ J then according to Theorem 1.2 the semi-trajectory Σ

can be extended to the

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Method of Lyapunov functions 189

left, i.e. there exists a continuous mapping ϕ : S → J such that π

ϕ(s) = ϕ(t + s)

for all t ∈ T

, s ∈ S and ϕ(0) = x. Since J is compact, the set of α–limit points

of the motion ϕ is not empty, compact and invariant. Obviously, α

∩ Θ 6= ∅

and, consequently, there exists t

→ −∞ such that |ϕ(t

)| → 0. Let ε > 0 and

δ(ε) > 0 be chosen out of the condition of the uniform stability of Θ. Then for n

big enough we have |ϕ(t

)| < δ and, consequently, |π

ϕ(t

)| = |ϕ(t + t

)| < ε for

all t ≥ 0. In particular, |x| = |ϕ(t

− t

)| < ε. And since ε is arbitrary, we have

|x| = 0, i.e. J ⊆ Θ.

We will show now that for every r > 0 the following equality takes place

lim

t→+∞

sup

|x|≤r

|xt| = 0. (5.31)

Suppose that it is not so; then there exist ε

> 0, r

> 0, |x

| < r

and t

→ +∞

such that

| ≥ ε

. (5.32)

In virtue of dissipativity of the system



(X, T

, π), (Y, T

, σ), h



the sequence {x

}

can be considered convergent. Assume x

:= lim

n→+∞

and note that x

∈ J,

hence |x

| = 0. The last contradicts to (5.32). So, the equality (5.31) takes place.

The continuity of V is established literally repeating the reasoning of Theorem 5.1

when instead of (5.2) we use the condition (5.31).

As for the 3rd condition, we verify it in the same way that in Theorem 5.1. The

theorem is proved. 

Note that Theorem 5.13 is valid if we replace X by some tubular neighborhood

:= {x ∈ X : |x| ≤ r} (r > 0) of the null section Θ.

Deﬁnition 5.5 The null section Θ is called uniformly asymptotically stable if it

is uniformly stable and there exists r > 0 such that lim

t→+∞

|xt| = 0 for all x ∈ U

If we slightly change the proof of Theorem 5.13, we can prove the following

theorem.

Theorem 5.14 Let Y be compact. For the null section Θ to be uniformly asymp-

totically stable it is necessary and suﬃcient that there would exist r > 0 and a

continuous function V : U

→ R

satisfying the following conditions:

(1) V (x) ≥ a(|x|) for all x ∈ U

, V (θ

) = 0 for all y ∈ Y ;

(2) V (xt) ≤ V (x) if xτ ∈ U

for all τ ∈ [0, t];

(3) the lines of level of V do not contain non-null ω-limit points of the dynamical

system (X, T

, π).

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190 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Remark 5.6 As well as in Lemma 5.1 we can show that in Theorem 5.14 the

condition 3. can be replaced by the following one: the lines of level of the function

V do not positive semi-trajectories of the dynamical system (X, T

, π).

At last note that the following theorem takes place.

Theorem 5.15 For the null section Θ would be uniformly stable it is necessary

and suﬃcient that there would exist r > 0 and a function V : U

→ R

satisfying

the following conditions:

1. V (x) ≥ a(|x|) for all x ∈ U

and lim

|x|→0

V (x) = 0;

2. V (xt) ≤ V (x) if xτ ∈ U

for all τ ∈ [0, t].

Proof. Suﬃciency. Let us show that the null section is uniformly stable. Suppose

that it is not so. Then there exist ε

> 0 (ε

< r), δ

↓ 0, |x

| < δ

and t

≥ 0

such that x

τ ∈ U

for all τ ∈ [0, t

] and

| ≥ ε

Note that

a(ε

) ≤ a(|x

|) ≤ V (x

) ≤ V (x

). (5.33)

Passing to limit in (5.33) as n → +∞, we obtain a(ε

) ≤ 0 and, consequently,

= 0. The last contradicts to the choice of ε

. So, the uniform stability of Θ is

proved.

Necessity. Let Θ be uniformly stable. For ε

= 1 select δ

= δ(ε

) > 0 (δ

≤ 1)

out of the condition of the uniform stability of Θ. Assume r := δ

and deﬁne

V : U

→ R

by the equality (5.1). It is easy to see that the function V is the one

in question, i.e. it satisﬁes the conditions 1. and 2. of Theorem 5.15. The theorem

is proved. 

Remark 5.7 By the conditions of Theorem 5.15 the function V , generally speak-

ing, is not continuous, unlike the case in Theorems 5.13 and 5.14.

From the theorems given in this section we can get the corresponding statements

for the equation (5.9). Let us previously bring the following deﬁnitions.

Let f(t, 0) ≡ 0 and the function f ∈ C(R ×E

, E

) be regular.

Deﬁnition 5.6 We will say that the null solution of the equation (5.9) is uniformly

stable if for every ε > 0 there exists δ(ε) > 0 such that g ∈ H(f), |v| < δ implies

|ϕ(t, v, g)| < ε for all t ≥ 0.

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Method of Lyapunov functions 191

Deﬁnition 5.7 If the null solution of the equation (5.9) is uniformly stable and

there exists γ > 0 such that lim

t→+∞

|ϕ(t, v, g)| = 0 for all g ∈ H(f) and v ∈ B(0, γ),

then the null solution of the equation (5.9) is called uniformly asymptotically stable.

Deﬁnition 5.8 The null solution of (5.9) is globally uniformly asymptotically

stable if it is uniformly stable and |ϕ(t, v, g)| → 0 as t → +∞ for any g ∈ H(f) and

v ∈ E

Note that from the results of the works

[

]

[

291

]

follows the equivalence of the

standard deﬁnition of the uniform stability (global uniform asymptotic stability)

and of the one given above.

From Theorems 5.13, 5.14 and Remark 5.6 follows the theorems below.

Theorem 5.16 Let H(f) be compact and f(t, 0) ≡ 0. For the null solution of the

equation (5.9) to be globally uniformly stable it is necessary and suﬃcient that there

would exists a continuous function V : H(f) × E

→ R

satisfying the following

conditions:

(1) V (g, v) ≥ a(|v|) for all v ∈ E

, V (g, 0) = 0, g ∈ H(f) and Im a = Im V where

a ∈ A;

(2) V (g

, ϕ(τ, v, g)) ≤ V (g, v) for all g ∈ H(f), v ∈ E

and τ ≥ 0;

(3) whatever would be the function g ∈ H(f), the lines of level of the function V do

not contain positive semi-trajectories of the equation (5.10) excepting the trivial

one.

Theorem 5.17 Let H(f) be compact and f(t, 0) ≡ 0. For the null solution of the

equation (5.9) to be uniformly asymptotically stable it is necessary and suﬃcient

that there would exist a continuous function V : H(f ) × B[0, r

] → R

satisfying

the following conditions:

(1) V (g, v) ≥ a(|v|) and V (g, 0) = 0 for all g ∈ H(f) and x ∈ B[0, r

], where a ∈ A;

(2) V (g

, ϕ(t, v, g)) ≤ V (g, v) if ϕ(t, v, g) ∈ B[0, r

] for all τ ∈ [0, t];

(3) whatever would be the function g ∈ H(f), the lines of level of the function V do

not contain positive semi-trajectories of the equation (5.10) excepting the trivial

one.

From Theorems 5.16 and 5.17 we can get the suﬃcient conditions of asymptotic

stability suitable for the applications.

Theorem 5.18 Let H(f ) be compact, f(t, 0) ≡ 0 and let there be a continuously

diﬀerentiable function V ∈ C(R × E

, R

) satisfying the conditions:

(1) a(|u|) ≤ V (t, u) ≤ b(|u|) (a, b ∈ A, Im a = Im b) for all t ∈ R and u ∈ E

;

(2) M

⊆ M

∩ M ∂V

∂t

∩ M

grad

;

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192 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(3)

(t, u) :=

∂

∂t

V (t, u) +



grad

V, f



≤ 0 (t ∈ R, u ∈ E

);

(4) whatever would be the function g ∈ H(f), the lines of level of the function V do

not contain positive semi-trajectories of the equation (5.10) excepting the trivial

one.

Then the null solution of the equation (5.9) is uniformly asymptotically stable in

general.

Theorem 5.19 Let r > 0, H(f ) be compact, f (t, 0) ≡ 0 and let exist a con-

tinuously diﬀerentiable function V ∈ C(R × B[0, r], R

) satisfying the following

conditions:

(1) a(|u|) ≤ V (t, u) ≤ b(|u|) (a, b ∈ A) for all t ∈ R and x ∈ B[0, r];

(2) M

⊆ M

∩ M ∂V

∂t

∩ M

grad

;

(3)

(t, u) :=

∂

∂t

V (t, u) +



grad

V (t, u), f (t, u)



≤ 0 (t ∈ R, u ∈ E

);

(4) whatever would be the function g ∈ H(f), the lines of level of the function V do

not contain positive semi-trajectories of the equation (5.10) excepting the trivial

one.

Then the null solution of the equation (5.9) is uniformly asymptotically stable.

Proof. The proof of Theorems 5.18 and 5.19 is constructed after the same pattern

as that in Theorem 5.9. 

Statements similar to Theorem 5.19 have been obtained in the works

[

]

[

193

]

5.4 Equations with convergence

1. Let us give the criterions of the convergence of the nonlinear equation (5.9) re-

sulting from Theorems 2.7, 2.12 and 2.13, Corollary 2.2 and Theorems 2.16 and 2.17

by applying the last theorems to the non-autonomous dynamical system generated

by the equation (5.9).

Theorem 5.20 Let f ∈ C(R×E

, E

) be recurrent w.r.t. t ∈ R uniformly w.r.t.

u on compacts from E

[

300

]

. For the equation (5.9) to be convergent it is necessary

and suﬃcient that the following conditions would be satisﬁed:

(1) the equation (5.9) has at least one bounded on R

solution;

(2) lim

t→+∞

|ϕ(t, v

, g) −ϕ(t, v

, g)| = 0 for all g ∈ H(f) and v

, v

∈ E

;

(3) for every ε > 0 and r > 0 there exists δ = δ(ε, r) > 0 such that |v

− v

| < δ

(|v

|, |v

| < δ) implies |ϕ(t, v

, g) −ϕ(t, v

, g)| < ε for all t ≥ 0 and g ∈ H(f).

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Method of Lyapunov functions 193

Note that in the case of almost periodicity of the function f Theorem 5.20

generalizes and reﬁnes the criterion of almost periodic convergence of V.I.Zubov

[

336

]

Theorem 5.21 Let f ∈ C(R×E

, E

) be recurrent w.r.t. t ∈ R uniformly w.r.t.

u on compacts from E

. For the equation (5.9) to be convergent it is necessary and

suﬃcient that the following conditions would be fulﬁlled:

1) the equation (5.9) has at least one bounded on R

solution;

2) whatever would be g ∈ H(f) and v ∈ E

, the solution ϕ(·, v, g) of the equation

(5.10) is asymptotically stable.

In the case of periodicity of the function f Theorem 5.21 coincides with the

theorem of N.N.Krasovskii-V.A.Pliss

[

270

]

Theorem 5.22 Let f ∈ C(R×E

, E

) be recurrent w.r.t. t ∈ R uniformly w.r.t.

u on compacts from E

and the equation (5.9) be dissipative. For the equation (5.9)

to be convergent it is necessary and suﬃcient that there would exists a continuous

function V : H(f ) ×E

×E

→ R

satisfying the following conditions:

1) V is positively deﬁned, i.e. V (g, v

, v

) = 0 (g ∈ H(f) and v

, v

∈ E

) if and

only if v

= v

;

2) V (g

, ϕ(t, v

, g), ϕ(t, v

, g)) ≤ V (g, v

, v

) for all t ≥ 0, v

, v

∈ E

and g ∈

H(f);

3) V (g

, ϕ(t, v

, g), ϕ(t, v

, g)) = V (g, v

, v

) for all t ≥ 0 if and only if v

, v

∈ E

In the case of the periodicity of the function f Theorem 5.22 coincides with

Theorems 7.2 and 7.3 from

[

270

]

Theorem 5.23 Let f ∈ C(R × E

, E

) be recurrent w.r.t. t ∈ R uniformly

w.r.t. u on compacts from E

and let the equation (5.9) be dissipative. For the

equation (5.9) to be convergent it is necessary and suﬃcient that there would exist a

continuous function V : H(f) ×E

×E

→ R

satisfying the following conditions:

(1) V is positively deﬁned;

(2) V (g

, ϕ(t, v

, g), ϕ(t, v

, g)) < V (g, v

, v

) for all t > 0, g ∈ H(f) and v

, v

∈

, v

6= v

;

2. Let E be a Banach space with the norm | · | and [E] be the Banach space of

all linear bounded operators acting from E to E and equipped with the operational

norm.

Consider linear non-homogeneous equation

˙u = A(t)u + f(t), (5.34)

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194 Global Attractors of Non-autonomous Dissipative Dynamical Systems

where A ∈ C(R, [E]) and f ∈ C(R, E). Along with the equation (5.34) we will

consider the corresponding homogeneous equation

˙u = A(t)u (5.35)

and denote by U(t, A) the Cauchy operator of equation (5.35).

In this paragraph we will suppose that the sets H(A) and H(f) are compact

in C(R, [E]) and C(R, E) respectively, though some of the statements given below

take place even without this condition.

Lemma 5.2 For the equation (5.34) to be convergent it is necessary and suﬃcient

that there would exist positive numbers N and ν such that

kU(t, A)U

−1

(τ, A)k ≤ N exp

−ν(t−τ)

(5.36)

for all t ≥ τ (t, τ ∈ R).

Proof. The formulated statement follows from the deﬁnition of convergence and

from the fact that uniform asymptotic stability of the null solution of the equation

(5.35) is equivalent to the condition (5.36). 

Lemma 5.3 The mapping U : R × C(R, [E]) → [E] (U : (t, A) → U(t, A)) is

continuous w.r.t. A uniformly w.r.t. t ∈ R on compacts from R, i.e. for every

` > 0 lim

n→+∞

max

|t|≤`

kU(t, A

) − U(t, A)k = 0 if A

→ A in C(R, [E]).

Proof. Let A

⊂ C(R, [E]), A

→ A be uniform on compacts from R and ` > 0.

Then there exists positive number M(`) such that

max

|t|≤`

(t)k ≤ M(`). (5.37)

Since U(t, A

) is the solution of the system

(

U(t, A

) = A

(t)U(t, A

)

U(0, A

) = Id

where Id

is a unit operator in E, then we have

kU(t, A

)k ≤ exp(

(s)kds) (t > t

, t, t

∈ [−`, `]).

From the last inequality follows that

max

|t|≤`

kU(t, A

)k ≤ exp(2`M(`)) (n = 1, 2, . . . ).

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Method of Lyapunov functions 195

For every n ∈ N the function V

(t) := U(t, A) −U(t, A

) satisﬁes the system

(

(t) = A(t)V

(t) + [A(t) − A

(t)]U(t, A

)

(0) = 0.

From this follows that

(t) = U(t, A)

−1

(τ, A)[A

(τ) − A(τ)]U (τ, A

)dτ. (5.38)

Let K(`) := max

|t|≤`

{kU(t, A)k, kU

−1

(t, A)k}. From (5.37) and (5.38) follows the in-

equality

max

|t|≤`

(t)k ≤ K

(l)2` exp(2`M(`)) max

|t|≤`

(t) − A(t)k. (5.39)

Passing to limit in (5.39) as n → +∞, we obtain

lim

n→+∞

max

|t|≤l

kU(t, A) − U (t, A

)k = 0.

The lemma is proved. 

Lemma 5.4 Let N > 0 and ν > 0 be such that there holds the inequality (5.36).

Then for every B ∈ H(A), t ≥ τ and τ ∈ R, the inequality

kU(t, B)U

−1

(τ, B)k ≤ N exp(−ν(t − τ )), (5.40)

holds where U(t, B) is the Cauchy operator of the equation

˙y = B(t)y (B ∈ H(A)).

Proof. Let N > 0 and ν > 0 be such that there takes place the inequality (5.36)

and let B ∈ H(A). Then there exists {τ

} ⊂ R such that A

→ B in C(R, [E]).

Note that U(t + τ, A) = U(t, A

)U(τ, A) for all t, τ ∈ R and A ∈ C(R, [E]), hence

U(t, A

−1

(τ, A

) = U(t + τ

, A)U

−1

(t + τ

, A). (5.41)

From what follows that

kU(t, A

−1

(τ, A

) ≤ N exp(−ν(t − τ)) (t ≥ τ, n = 1, 2, . . . ).

Passing to limit in the last inequality as n → +∞ and taking into account Lemma

5.3 we will obtain the inequality (5.40). The lemma is proved. 

Lemma 5.5 Let the condition (5.36) be fulﬁlled. Then for every B ∈ H(A) and

p ≥ 1 by the equality

kvk

B,p

= {

+∞

|U(t, B)v|

dt}

(5.42)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 5

196 Global Attractors of Non-autonomous Dissipative Dynamical Systems

on E is deﬁned a norm that is topologically equivalent to the old and moreover

kU(τ, B)v

−U(τ, B)v

≤ N(

)

exp(−ντ)kv

−v

B,p

(5.43)

for all τ ≥ 0.

Proof. From the inequality of Minkovskii follows that by the equality (5.42) is

deﬁned some norm on E. Let us show that there exist positive number m

and M

that does not depend on B ∈ H(A) such that m

|v| ≤ kvk

B,p

≤ M

|v|. The ﬁrst

inequality follows from (5.36) and from the inequality

kvk

B,p

+∞

|U(t, B)v|

dt ≤

+∞

kU(t, B)k

dt|v|

≤

+∞

exp(−νpt)dt|v|

νp

|v|

On the other hand since U(−τ, B

)U(τ, B)v = v for all τ ∈ R and v ∈ E, then

|v| = |U(−τ, B

)U(τ, B)v| ≤ kU(−τ, B

)k|U(τ, B)v| ≤

exp(a|τ|)|U (τ, B)v| (a := sup{kA(t)k : t ∈ R} < +∞)

and, consequently,

kvk

B,p

+∞

|U(t, B)v|

dt ≥

+∞

(exp(−at)|v|)

dt =

+∞

exp(−apt)dt|v|

|v|

= (ap)

−

). So,

(ap)

−

|v| ≤ kvk

B,p

≤ N(νp)

−

|v|.

Now let us establish the inequality (5.43). For that we will note that

kU(τ, B)v

−U(τ, B)v

+∞

|U(t, B

)[U(τ, B)v

−U(τ, B)v

+∞

|U(t, B

)U(τ, B)(v

−v

dt =

+∞

|U(t + τ, B)(v

−v

+∞

|U(s, B)(v

−v

ds ≤

+∞

exp(−νps)|v

−v

νp

exp(−νpτ)|v

−v

≤

νp

exp(−νpτ)kv

−v

B,p

From the last inequality follows (5.43). The lemma is proved. 